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Problem Set 4 Solutions
Chemistry 4531
−
iE1 t
−
iE2 t
+ ψ 2(x) e
1.
We showed in class that the function Ψ s ( x, t ) = ψ 1(x) e
is a solution to the time-dependent Schrödinger equation for a system with
Hamiltonian H and eigenfunctions ψ1 and ψ2 corresponding to eigenenergies E1 and
E2. Simplify the expression for Ψs*Ψs and then obtain an expression for the beat
frequency between ψ1 and ψ2 in terms of h, E1 and E2.
Ψ s ( x, t ) Ψ s ( x, t ) = ψ 1(x) e
= ψ *1ψ 1 + ψ *2ψ 2 + ψ *1e
= 2+ ψ1 ψ 2 e
i ( E1 − E2 ) t
= 2+2 ψ1 ψ 2 e
iE1 t
−
iE1 t
ψ 2e
+ ψ 2(x) e
−
iE2 t
+ ψ 1 ψ 2 *e
−
iE2 t
+ψ *2 e
iE2 t
ψ 1(x) e
ψ 1e
−
−
iE1 t
+ ψ 2(x) e
−
iE2 t
iE1 t
i ( E1 − E2 ) t
i ( E1 − E2 ) t
The coefficient of the complex exponential is ωt = 2πνt, so the beat frequency
ν = (E1-E2)/h, as we expected
1
2.
Consider three non-interacting particles of masses M, 2M and 4M,
constrained to lie in a square with sides of length L. How many quantum numbers
are there for this system? Separate variables and obtain the eigenfunctions and
eigenvalues. What are the degeneracies of the three lowest energy levels?
There will be 6 quantum numbers.
Call the three particles 1,2,3 respectively.
Incorporate the relative mass into the quantum number part of the expression.
2
2
⎛ 2
nx22 n y 2 nx23 n y 3 ⎞
2
+
+
+
En x , n y , n z =
⎜ nx1 + n y1 +
⎟ n x1 = 1, 2,… , n y1 = 1, 2,… , n x2 = 1, 2,…
2 ML2 ⎜⎝
2
2
4
4 ⎟⎠
The wave function is a product of six particle in a box eigenfunctions, as you know.
π2
2
Let us express all energies in units of
π2 2
.
2ML2
The lowest level is (1,1,1,1,1,1,) with energy 3.5. It is non degenerate (or singly degenerate, if
you prefer).
Second is (1,1,1,1,2,1) with energy 4.25. It is twofold degenerate. The (1,1,1,1,1,2) state has
the same energy.
Third is (1,1,1,1,2,2) with energy 5. Other levels with the same energy are (1,1,2,1,1,1) and
(1,1,1,2,1,1).
Thus the third energy level is three-fold degenerate.
Going higher in energy,
(111113) and (111131) are at energy 5.5
and ((111123) and (111132 are at energy 6.25
2
3.
Find the wavefunctions and energy levels for a particle of mass M contained
in a cube with sides of length L. What is the degeneracy of the level with energy
three times greater than that of the ground state?
En
x
ψn
,ny ,nz
x
=
2
(n
2ML
2
2
x
+ ny2 + nz2
nx = 1,2,…, ny = 1,2,…, nz = 1,2,…
⎛ ny π y
⎛n πx ⎞
sin ⎜ x
⎟ sin ⎜⎜
⎝ ⎠
⎝ L ⎠
⎝ L
( x , y , z ) = ⎜ 2L ⎟
⎛ ⎞
,ny ,nz
3
2
)
⎞
⎛ n πz ⎞
⎟ sin ⎜ z
⎟ inside the cube, and zero outside
⎟
L
⎝
⎠
⎠
2
The ground state (1,1,1) has energy 3 in units of
2ML2
. Thus we wish to find which levels
have energy 9?
The answer is (2,2,1), (2,1,2) and (1,2,2). Any quantum number >2 will exceed the value 9 for
the total energy. The level is said to be threefold degenerate
3